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Units divided by Acre

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The Future Value of Stephania's college fund will be =$106,817.02. To save this amount over a period of 18 years or 72 quarters @ 6.4% compounded quarterly, they will need to invest =$787.60 at the beginning of each quarter for 18 years or 72 quarters.

P.S. Do you know or understand the TVM formulas used to calculate these amounts? If you don't, then let us know here and will explain them to you.

tan( 30º )  =  25 / x               Multiply both sides of the equation by  x  .

x tan( 30º )  =  25                 Divide both sides of the equation by tan( 30º )  .

x  =  25 / tan( 30º )               And   tan( 30º ) = √3 / 3

x  =  25 / ( √3 / 3 )

x  =  25 * 3 / √3

x  =  75 / √3                         Multiply numerator and denominator by  √3  .

x  =  75√3 / 3                       Reduce fraction by  3 .

x  =  25√3

I remember this question from a while back....

The principal, P,  must have been  

Ending Amount - Interest Earned  =

 $6044.34 -  $1344.34  =  $4700

We can find the interest rate, r, thusly :

 $6044.34 =  $4700 ( 1 + r/12)^(12 * 6)

$6044.34 =  $4700 ( 1 + r/12)^(72 )           divide both sides by $4700

6044.34 /  4700  =  ( 1 + r/12)^(72 )     take the log of both sides

log ( 6044.34 /  4700 )  = log ( 1 + r/12)^(72 )   and we can write

log ( 6044.34 /  4700 )  = 72 * log ( 1 + r/12)       divide both sides by 72

log ( 6044.34 /  4700 )  / 72  =   log ( 1 + r/12)

0.00151738  =  log ( 1 + r/12)     and we can write

10^ (0.00151738)  =  ( 1 + r/12)  subtract 1 from both sides

10^ (0.00151738) - 1  =  r/12  

0.00350001 =  r/12     multiply both sides by 12

.042  =  r  =     4.2%

$6,044.34 - $1,344.34 =$4,700 amount invested.

$6,044.34/4,700 =1.28603.......

1.28603^(1/72) =1.0035

1.0035 -1 =0.0035 Interest rate per month.

0.0035 x 1200 = 4.20% annual interest rate compounded monthly.

h(x)  =  -343^x 

The domain is all real numbers

The range  is   ( - infinity, 0 )

As x decreases, h(x)  increases to the upper bound of ≈   0

As x increases, h (x)  decreases ......the right end behavior is towards   - infinity

Here's the graph : https://www.desmos.com/calculator/g85xk6uq8p

Mmmmm....this one looks interesting  !!!

We can let the amount after year 1 be represented as

P ( 1 + r)  = 2251.05     (1)

Where P is the orginal amount invested  and r is the interest rate

After year 2 we have

P (1 + r)^2  =  2356.85    (2)

Dividing  (2) by (1)  we have that

(1 + r)  = 2356.85 /  2251.05       subtract  1 from both sides

r  =    2356.85 /  2251.05    - 1    ≈  .047 ≈  4.7 %

We can find the original amount invested as follows

P  ( 1 + .047)  =  2251.05     divide both sides by  (1 + .047)

P  =  2251.05 / ( 1 + .047 =  $2150

We can find the time to double - the amount will be 2P - with this :

log [ (2P / P) ]  /  log ( 1 + interest rate )

log (2)  / log (1+. 047)  ≈  15.09  years

Year 4 / Year 1 =2,583.60/2,251.05 =1.147731...

1.147731^(1/3) =4.7% this the yearly compound rate.

2,251.05 / 1.047=$2,150 - the amount that Stacy invested.

 2,150 x 1.047^n =4,300.

n = 15.09 years to double the money.

However....I'm sure  that the answer isn't 450, either....

Assuming that we can't begin with 0, there are 900 integers from 100 - 999 and half of these will be even = 450....and some of those will have repeated digits.....so.....the answer is less than 450

Here's a "brute force"  method

Let's look at all the integers that end in 0

There are  10 of these in each group of 100.....and we have 9 groups of 100 from 100-999

So....that's 10 * 9  = 90 possible integers.....but...we have to discard 10 of these (100, 200, 300. etc.) because of the repeated 0....so......that makes 80 possible numbers ending in 0

Now.....let's look at all the numbers ending in 2

Again there are 10 of these in each group of 100.....so, again, that's 90 possible numbers

But.....from these we must subtract all the numbers that have 2 as a middle digit and 2 as an ending digit.....and there are 9 of these   (122, 222, 322, 422, etc )...so that gives us 81 possible numbers....

Also.....we have to subtract the numbers that start with 2 and have 2 as an ending digit....there are 9 of these, as well  (202, 212, 232, 242, etc. ).....(note that we have already counted '222")....so.....the possible numbers ending in 2 with no repeated digits  = 90 - 9 - 9  =  72

And the other numbers ending in 4,6 or 8  will similarly occur 72 tmes each

So.....the total combinations ending in an even number with no repeated digits  = 80 +  4 * 72  =

80 + 288  =   368

As Melody always says, "That's what I think!!! "

Haha you're not dumb !! Glad you got it now! 

OHHHHHHHHkay thanks I'm dumb lelel

What do you mean by switch y and x?

The point is still

(x , y)

But since y =  f(x) , we can replace  " y "  with  " f(x) "

So the point can also be written as

(x , f(x) )

(x , y)  =  (x , f(x) )       if            y = f(x)

I didn't know if that was a confusing part or not. 

Hm. Okay, thanks. That helps. I'm not sure why we switch y with x but I think I'll be able to remember that it's what we do anyways so thanks! :)

If we get the equation into this form:   y = mx + b     ,  " m "  will be the slope.

5y  =  x - 3                       Multiply through by \(\frac15\) .

\(\frac55y=\frac15x-\frac35 \\~\\ y=\frac15x-\frac35\)

Now we can see that  " m "  is    \(\frac15\)     , the slope.

If  y = f(x)  , we can say this will pass through the point

( x , f(x) )                And they tell us that when  x = -1/2  ,  f(x) = -2/3

( -1/2 , -2/3 )

If  y = | f(x) |  , we can say this will pass through the point

( x , | f(x) | )             And they tell us that when x = -1/2  ,  f(x) = -2/3

( -1/2 , | -2/3 | )

( -1/2 , 2/3 )

I hope this made some sense! 

This is very easy and simple way to solve such problems:

30% means =30/100 =0.30, then:

0.30 x 5,600 =1,680  and that is it.

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